A240559 -id:A240559 - cp's OEIS frontend

A240561 The main diagonal in the difference table of A240559.

0, 1, -10, 178, -5296, 238816, -15214480, 1301989648, -144118832896, 20040052293376, -3419989086092800, 702831038438522368, -171209091176316215296, 48783404012394865985536, -16074763418934659189278720, 6065554251200571899397081088, -2598468976240882751482797162496

Offset: 0

Peter Luschny, Apr 17 2014

sign

			a(n) is the main diagonal in this difference table D(n, k):
[    0,     0,     1,    -3,    -5,    45,    61, -1113, -1385]
[    0,     1,    -2,    -8,    40,   106, -1052, -2498]
[    1,    -1,   -10,    32,   146,  -946, -3550]
[    0,   -11,    22,   178,  -800, -4496]
[  -11,    11,   200,  -622, -5296]
[    0,   211,  -422, -5918]
[  211,  -211, -6340]
[    0, -6551]
[-6551]
D(n, 0) = A240560(n).
D(0, n) = A240559(n).
D(2*n, 0) = (-1)^(n+1)*A147315(2*n, 2).

Cf. A147315, A240559, A240560.

A240561_list := proc(len) local A, m, n, k;
n := 2*len-1; A := array(0..n, 0..n);
for m from 0 to n do
   A[m, 0] := euler(m) + 2^(m+1)*euler(m+1,0);
   for k from m-1 by -1 to 0 do
      A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1]
od od; [seq(A[k, k], k=0..len-1)] end:
A240561_list(17);

Table[-Sum[Binomial[n, k]*EulerE[n+k+1], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Apr 06 2015 *)

a(n):=-sum(binomial(n,k)*euler(n+k+1),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

a(n) = -Sum_{k=0..n}(C(n,k)*Euler(n+k+1)). - Vladimir Kruchinin, Apr 06 2015

a(n) ~ (-1)^(n+1) * 2^(4*n+9/2) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Apr 06 2015

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

Original entry on oeis.org

0, -3, 45, -1113, 42585, -2348973, 176992725, -17487754833, 2195014332465, -341282303124693, 64397376340013805, -14499110277050234553, 3840151029102915908745, -1182008039799685905580413, 418424709061213506712209285, -168805428822414120140493978273

Offset: 0

Peter Luschny, Apr 17 2014

sign

Cf. A000182, A000364, A028296, A240559, A147315, A186370.

A241242 := proc(n) e := n -> euler(n,1/2) + euler(n,1); -2^(2*n+1)*(e(2*n+1) + 2*e(2*n+2)) end: seq(A241242(n),n=0..15);

Array[-2^(2 # + 1)*(EulerE[2 # + 1, 1/2] + EulerE[2 # + 1, 1] + 2 (EulerE[2 # + 2, 1/2] + EulerE[2 # + 2, 1])) &, 16, 0] (* Michael De Vlieger, May 24 2018 *)

a(n) = A240559(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1).
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*2^k*(E(k, 1/2) + 2*E(k+1, 0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*(skp(k, 0) + skp(k+1, -1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = Bernoulli(2*n + 2) * 4^(n+1) * (1 - 4^(n+1)) / (2*n + 2) - EulerE(2*n + 2), where EulerE(2*n) is A028296. - Daniel Suteu, May 22 2018
a(n) = (-1)^(n+1) * (A000182(n+1) - A000364(n+1)). - Daniel Suteu, Jun 23 2018

A240560 a(n) = 2^nE(n,1/2) + 2^(n+1)E(n+1,0), where E(n,x) the Euler polynomials.

Original entry on oeis.org

0, 0, 1, 0, -11, 0, 211, 0, -6551, 0, 303271, 0, -19665491, 0, 1704396331, 0, -190473830831, 0, 26684005437391, 0, -4581126864886571, 0, 946075012113714451, 0, -231406946026650896711, 0, 66164529094650835995511, 0, -21866924546405967976005251

Offset: 0

Peter Luschny, Apr 17 2014

sign

Cf. A122045, A155585, A240559, A240561.

A240560 := n -> euler(n) + 2^(n+1)*euler(n+1, 0):
seq(A240560(n), n=0..28);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*If[n==k, 1, x^(n-k)], {k, 0, n}];
a[n_] := skp[n, 0] + skp[n+1, -1];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 08 2019 *)

a(n) = skp(n, 0) + skp(n+1, -1), where skp(n, x) are the Swiss-Knife polynomials A153641.

a(n) = A122045(n) - A155585(n+1).

cp's OEIS Frontend

A240561 The main diagonal in the difference table of A240559.

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A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

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Formula

A240560 a(n) = 2^nE(n,1/2) + 2^(n+1)E(n+1,0), where E(n,x) the Euler polynomials.

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cp's OEIS Frontend

A240561 The main diagonal in the difference table of A240559.

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Author

Keywords

Examples

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Maple

Mathematica

Maxima

Formula

A241242 a(n) = -2^(2*n+1)*(E(2*n+1, 1/2) + E(2*n+1, 1) + 2*(E(2*n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

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Maple

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Formula

A240560 a(n) = 2^n*E(n,1/2) + 2^(n+1)*E(n+1,0), where E(n,x) the Euler polynomials.

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Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

A240560 a(n) = 2^nE(n,1/2) + 2^(n+1)E(n+1,0), where E(n,x) the Euler polynomials.